Being interested in the Philippines up to 1942, I've seen a number of POD-marked stocks - nearly all of them had "PODD", the last initial, the inspector. Virtually all of the stocks also had Benecia Arsenal, in California - either the rifles were shipped through that arsenal or they came back through that arsenal.

Yes, you have an interesting stock - I see a very early "script" inspection stamp - 1910 or earlier, then the 1918 RIA stamp and then the POD and BAWL. More than likely, your rifle saw service up through WWI and then was overhauled after WWI, possibly at Rock Island. The stock was probably taken off an earlier rifle. Whether some of the stamps are "from" your rifle or the stock was added by a previously civilian owner would be hard to say.

Oh, and all Remington M1903s were "Modified", not just the later ones. (Sorry, pet peeve!!)